Problem 16 - Algebra for College Students

Question

A disc jockey must choose 8 songs from the top 40 to play in the next 30 -minute segment of his show. How many different arrangements are possible for this segment?

Step-by-Step Solution

Answer

There are 121,080,960,000 possible arrangements.

Step 1: Understand the problem

We need to determine the number of different arrangements of 8 songs that the disc jockey can choose from a list of 40 songs.

Step 2: Use the permutation formula

Since the order in which the songs are played matters, we use the permutation formula. The number of permutations of 8 items from a set of 40 is given by the formula: \[ P(n, k) = \frac{n!}{(n-k)!} \]where \( n \) is the total number of items to select from (40), and \( k \) is the number of items to arrange (8).

Step 3: Substitute values into the permutation formula

Substitute \( n = 40 \) and \( k = 8 \) into the permutation formula: \[ P(40, 8) = \frac{40!}{(40-8)!} = \frac{40!}{32!} \]

Step 4: Simplify the factorial expression

Simplify the expression by cancelling out \( 32! \) in the numerator and denominator: \[ P(40, 8) = 40 \times 39 \times 38 \times 37 \times 36 \times 35 \times 34 \times 33 \]Calculate the product of these numbers.

Step 5: Calculate the product

Calculate step-by-step: \[ 40 \times 39 \times 38 \times 37 \times 36 \times 35 \times 34 \times 33 = 121,080,960,000 \]

Key Concepts

Permutation Formula

In mathematics, permutations are a way to count the number of possible arrangements of a set of items. When the order of items matters, we use the permutation formula. The permutation formula is: \[ P(n, k) = \frac{n!}{(n-k)!} \] Here, \( n \) represents the total number of items to choose from, and \( k \) is the number of items we want to arrange. This formula helps to systematically count the total possible ways to arrange \( k \) items out of \( n \). For example, if a disc jockey wants to pick 8 songs out of a list of 40 and arrange them in a specific order, we would use this permutation formula to find the number of possible sequences.

Factorial

A factorial is a product of all positive integers up to a given number. It is denoted by an exclamation mark (!). For example, the factorial of 5, written as \( 5! \), is \( 5 \times 4 \times 3 \times 2 \times 1 = 120 \). In the permutation formula \( \frac{40!}{32!} \), we calculate the factorial of 40, which includes all numbers from 40 down to 1, and then divide by the factorial of 32, which includes numbers from 32 down to 1. Because \( 40! \) and \( 32! \) share the same factors from 32 to 1, these factors cancel each other out. What's left is the product of the numbers between 40 and 33. Factorials grow very quickly, making them useful for counting large sets of items.

Arrangement

An arrangement refers to the specific order in which items are placed. When choosing and arranging songs, the order is crucial because playing song A first and then song B is different from playing song B first and then song A. This is why we use permutations. For our disc jockey example, the total number of arrangements of 8 songs chosen from 40 is \( 40 \times 39 \times 38 \times 37 \times 36 \times 35 \times 34 \times 33 \). Each number multiplies by the next smaller number until we reach the 8th term, resulting in a total of 121,080,960,000 unique arrangements. Understanding this concept helps us grasp how order impacts the total possibilities.